×

zbMATH — the first resource for mathematics

Conditional problem for objective probability. (English) Zbl 1274.60013
Summary: The marginal problem consists in finding a joint distribution whose marginals are equal to the given less-dimensional distributions. Let’s generalize the problem so that there are given not only less-dimensional distributions but also conditional probabilities.
It is necessary to distinguish between objective (Kolmogorov) probability and subjective (de Finetti) approach. In the latter, the coherence problem incorporates both probabilities and conditional probabilities in a unified framework. Different algorithms available for its solution are described, e.g., in [A. Gilio and S. Ingrassia, “Geometrical aspects in checking coherence of probability assessments”, in: Proceedings of the 6th international conference on information processing and management of uncertainty in knowledge-based systems (IPMU’96), Granada, Spain. 55–59 (1996); G. Coletti and R. Scozzafava, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4, No. 2, 103–127 (1996; Zbl 1232.03010)]. In the context of the former approach, it is shown that it is possible to split the task into solving the marginal problem independently and to subsequent solving the pure conditional problem as certain type of optimization.
First, an algorithm (conditional problem) that generates a distribution whose conditional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heuristical. Due to the computational complexity, it is efficient for small size problems, e.g., five dichotomical variables. Secondly, a method is mentioned how to unite marginal and conditional problem to a more general consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for a limited number of variables and conditionals. The described approach makes it possible to integrate additional knowledge, contained, e.g., in an empirical distribution, in the solution of the consistency problem.

MSC:
60A99 Foundations of probability theory
65C50 Other computational problems in probability (MSC2010)
Citations:
Zbl 1232.03010
PDF BibTeX XML Cite
Full Text: Link
References:
[1] Cheeseman P.: A method of computing generalized Bayesian probability values of expert systems. Proceedings of the 6-th Joint Conference on Artificial Intelligence (IJCAI-83), Karlsruhe, pp. 198-202
[2] Deming W. E., Stephan F. F.: On a least square adjustment of sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 (1940), 427-444 · Zbl 0024.05502
[3] Gilio A., Ingrassia S.: Geometrical aspects in checking coherence of probability assessments. IPMU’96: Proceedings of the 6th International IPMU Conference (B. Bouchon-Meunier, M. Delgado, J. L. Verdegay, M. A. Vila, R. Yager, Granada 1996, pp. 55-59
[4] Coletti G., Scozzafava R.: Characterization of coherent conditional probabilities as a tool for their assessment and extension. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems, 4 (1996), 2, 103-127 · Zbl 1232.03010
[5] Kellerer H. G.: Verteilungsfunktionen mit gegebenen Marginalverteilungen. Z. Wahrsch. verw. Gebiete 3 (1964), 247-270 · Zbl 0126.34003
[6] Kříž O.: Invariant moves for constructing extensions of marginals. IPMU’94: Proceedings of the 5th International IPMU Conference (B. Bouchon-Meunier, R. Yager, Paris 1994, pp. 984-989
[7] Kříž O.: Optimizations on finite-dimensional distributions with fixed marginals. WUPES 94: Proceedings of the 3-rd Workshop on Uncertainty Processing (R. Jiroušek, Třeš\? 1994, pp. 143-156
[8] Kříž O.: Marginal problem on finite sets. IPMU’96: Proceedings of the 6-th International IPMU Conference (B. Bouchon-Meunier, M. Delgado, J. L. Verdegay, M. A. Vila, R. Yager, Granada 1996, Vol. II, pp. 763-768
[9] Kříž O.: Inconsistent marginal problem on finite sets. Distributions with Given Marginals and Moment Problems (J. Štěpán, V. Beneš, Kluwer Academic Publishers, Dordrecht - Boston - London 1997, pp. 235-242 · Zbl 0907.60003
[10] Scozzafava R.: A probabilistic background for the management of uncertainty in Artificial Intelligence. European J. Engineering Education 20 (1995), 3, 353-363
[11] Vicig P.: An algorithm for imprecise conditional probability assesment in expert systems. IPMU’96: Proceedings of the 6-th International IPMU Conference (B. Bouchon-Meunier, M. Delgado, J. L. Verdegay, M. A. Vila, R. Yager, Granada, 1996, Vol. I, pp. 61-66
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.